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%I #8 May 01 2018 03:00:18
%S 1,5,9,14,18,22,27,31,35,39,43,48,52,56,60,65,69,73,77,82,86,90,95,99,
%T 103,107,111,116,120,124,128,133,137,141,145,150,154,158,163,167,171,
%U 175,179,184,188,192,196,201,205,209,213,218,222,226,231,235,239
%N Solution (a(n)) of the system of 5 complementary equations in Comments.
%C Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3;:
%C a(n) = least new;
%C b(n) = least new;
%C c(n) = least new;
%C d(n) = least new;
%C e(n) = a(n) + b(n) + c(n) + d(n);
%C where "least new k" means the least positive integer not yet placed.
%C ***
%C Conjecture: for all n >= 0,
%C 0 <= 17n - 11 - 4 a(n) <= 4
%C 0 <= 17n - 7 - 4 b(n) <= 4
%C 0 <= 17n - 3 - 4 c(n) <= 3
%C 0 <= 17n + 1 - 4 d(n) <= 3
%C 0 <= 17n - 5 - e(n) <= 3
%C ***
%C The sequences a,b,c,d,e partition the positive integers. The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).
%H Clark Kimberling, <a href="/A299405/b299405.txt">Table of n, a(n) for n = 0..1000</a>
%e n: 0 1 2 3 4 5 6 7 8 9
%e a: 1 5 9 14 18 22 27 31 35 39
%e b: 2 6 11 15 19 23 28 32 36 40
%e c: 3 7 12 16 20 24 29 33 37 41
%e d: 4 8 13 17 21 25 30 34 38 42
%e e: 10 26 45 62 78 94 114 130 146 162
%t z = 200;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
%t Last[a] + Last[b] + Last[c] + Last[d]];
%t Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
%t AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
%t AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
%t AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
%t AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
%t Take[a, 100] (* A299405 *)
%t Take[b, 100] (* A299637 *)
%t Take[c, 100] (* A299638 *)
%t Take[d, 100] (* A299641 *)
%t Take[e, 100] (* A299409 *)
%Y Cf. A036554, A299634, A299637, A299638, A299641, A299409.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Apr 22 2018
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